Let $X=W_1^T\Sigma W_2$ be the singular value decomposition of $X$, where $W_1, W_2$ are unitaries and $\Sigma$ diagonal matrix. Taking $V=\sqrt{\Sigma}W_1$, $U=\sqrt{\Sigma}W_2$, the minimum is achieved.
I am not sure whether I understand your comment correctly. The other direction is easy, $\|X\|_1=\|V^TU\|_1\le \|V\|_2\|U\|_2\le (\|V\|_2^2+\|U\|_2^2)/2$, where $\|\cdot\|_1$, $\|\cdot\|_2$ denote trace norm, Frobenius norm, respectively. The first inequality follows from log majorization for singular values, the second one is by AM-GM inequality.